Calculus in Maths AI focuses on applications rather than technique. SL covers basic differentiation/integration for optimisation and area problems. HL extends to differential equations, coupled systems, slope fields, and numerical methods.
Derivative as rate of change. Power rule: d/dx(xⁿ) = nxⁿ⁻¹. Derivatives of polynomial, exponential, and trigonometric functions. Chain rule for composite functions. Finding gradients, tangent lines. Increasing/decreasing functions. SL focuses on interpretation rather than complex technique.
The primary SL application: finding maximum/minimum values in real contexts. Model the quantity to optimise as a function, differentiate, set f\'(x) = 0, and check with second derivative or sign analysis. Applications: minimising cost, maximising area, optimising dimensions of containers.
Anti-differentiation as reverse of differentiation. Definite integral as area under the curve. Trapezoidal rule for numerical approximation: ∫ₐᵇ f(x)dx ≈ (h/2)[f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)]. SL: simple polynomial integration. Kinematics: displacement from velocity, distance travelled.
Modelling with dy/dx = f(x,y). Slope fields: visualising solution curves. Euler\'s method: yₙ₊₁ = yₙ + h·f(xₙ,yₙ) for numerical approximation. Separable equations. Coupled differential equations (predator-prey models, SIR epidemic models). Phase portraits.
Yes, the calculus in Maths AI is significantly less demanding than in Maths AA. AI focuses on applications (optimisation, area, modelling with DEs) and uses technology heavily. AA requires more formal technique (chain rule, product rule, integration by parts, formal proofs). The volume of calculus is also smaller: ~15-20% of AI vs ~30-40% of AA. However, AI HL differential equations (coupled systems, Euler\'s method) can be conceptually challenging.
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